Families of varieties of general type. July 20, 2017

Transcription

1 Families of varieties of general type János Kollár July 20, 2017

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3 Contents Chapter 1. Introduction Short history of moduli problems From smooth curves to canonical models From stable curves to stable varieties Examples of bad moduli problems Compactifications of M g More unexpected examples Coarse and fine moduli spaces Singularities of stable varieties 50 Chapter 2. One-parameter families Locally stable families Locally stable families of surfaces Examples of locally stable families Stable families Cohomology of the structure sheaf Families of divisors I Boundary with coefficients > Grothendieck Lefschetz-type theorems Torsion in Grothendieck Lefschetz-type theorems 108 Chapter 3. Families of stable varieties Chow varieties and Hilbert schemes Representable properties Divisorial sheaves Local stability over reduced schemes Stability is representable I Moduli spaces of stable varieties I 143 Chapter 4. Families over reduced base schemes Statement of the main results Examples Families of divisors II Generically Q-Cartier divisors Stability is representable II Varieties marked with divisors Moduli of marked slc pairs I Stable families over smooth base schemes 186 Chapter 5. Numerical flatness and stability criteria 191 3

4 4 CONTENTS 5.1. Statements of the main theorems Simultaneous canonical models and modifications Examples Stability criteria in codimension Deformations of slc pairs Simultaneous canonical models Simultaneous canonical modifications Mostly flat families of line bundles Families over higher dimensional bases 222 Chapter 6. Infinitesimal deformations First order deformations with Klaus Altmann Deformations of cyclic quotient singularities with Klaus Altmann 238 Chapter 9. Hulls and Husks S 2 sheaves Hulls of coherent sheaves Relative hulls Universal hulls Husks of coherent sheaves Moduli space of quotient husks Hulls and Hilbert polynomials Moduli space of universal hulls Hulls and husks over algebraic spaces 271 Chapter 10. Ancillary results Flat families of S m sheaves Cohomology over non-proper schemes Dévissage Volumes and intersection numbers Double points Flatness criteria Noether normalization 305 Bibliography 311

5 CHAPTER 1 Introduction The moduli spaces of smooth projective curves of genus g 2, and their compactifications by the moduli space of stable projective curves of genus g are, quite possibly, the most studied of all algebraic varieties. The aim of this book is to generalize the moduli theory of curves to surfaces and to higher dimensional varieties. In the introduction we start to outline how this is done, and, more importantly, to explain why the answer for surfaces is much more complicated than for curves. On the positive side, once we get the moduli theory of surfaces right, the higher dimensional theory works the same. Section 1.1 is a quick review of the history of moduli problems, culminating in an outline of the basic moduli theory of curves. Section 1.2 introduces canonical models, which are the basic objects of moduli theory in higher dimensions. Starting from stable curves, Section 1.3 leads up to the definition of stable varieties, their higher dimensional analogs. Then we show, by a series of examples, why flat families of stable varieties are not the correct higher dimensional analogs of flat families of stable curves. Finding the correct replacement has been one of the main difficulties of the whole theory. Next we give a collection of examples showing how easy it is to end up with rather horrible moduli problems. Hypersurfaces are discussed in Section 1.4 and alternate compactification of the moduli of curves in Section 1.5. Further interesting examples are given in Section 1.6 while Section 1.7 illustrates the differences between fine and coarse moduli spaces. In Section 1.8 we recall the most important definitions and results about singularities that occur on stable varieties. An overview of the moduli theory of higher dimensional varieties is given in [Kol13b] Short history of moduli problems Let V be a reasonable class of objects in algebraic geometry, for instance, V could be all subvarieties of P n, all coherent sheaves on P n, all smooth curves or all projective varieties. The aim of the theory of moduli is to understand all reasonable families of objects in V and to construct an algebraic variety or scheme, or algebraic space whose points are in natural one-to-one correspondence with the objects in V. If such a variety exists, we call it the moduli space of V and denote it by M V. The simplest, classical examples are given by the theory of linear systems and families of linear systems. 1.1 Linear systems. Let X be a normal projective variety over an algebraically closed field k and L a line bundle on X. The corresponding linear system is LinSysX, L = {effective divisors D such that O X D = L}. 5

6 6 1. INTRODUCTION The objects in LinSysX, L are in natural one-to-one correspondence with the points of the projective space P H 0 X, L which is classically denoted by L. Thus, for every effective divisors D such that O X D = L there is a unique point [D] L. Moreover, this correspondence between divisors and points is given by a universal family of divisors over L. That is, there is an effective Cartier divisor Univ L L X with projection π : Univ L L such that π 1 [D] = D for every effective divisor D linearly equivalent to L, The classical literature never differentiates between the linear system as a set and the linear system as a projective space. There are, indeed, few reasons to distinguish them as long as we work over a fixed base field k. If, however, we pass to a field extension K k, the advantages of viewing L as a k-variety appear. For any K k, the set of effective divisors D defined over K such that O X D = L corresponds to the K-points of L. Thus the scheme theoretic version automatically gives the right answer over every field. 1.2 Jacobians of curves. Let C be a smooth projective curve or Riemann surface of genus g. As discovered by Abel and Jacobi, there is a variety Jac 0 C of dimension g whose points are in natural one-to-one correspondence with degree 0 line bundles on C. As before, the correspondence is given by a universal line bundle L univ C Jac 0 C, called the Poincaré bundle, That is, for any point p Jac 0 C, the restriction of L univ to C {p} is the degree 0 line bundle corresponding to p. A somewhat subtle point is that, unlike in 1.1, the universal line bundle L univ is not unique and need not exist if the base field is not algebraically closed. This has to do with the fact that while a divisor D X has no automorphisms fixing X, any line bundle L C has automorphisms that fix C: we can multiply every fiber of L by the same nonzero constant. 1.3 Chow varieties. Historically the next to emerge was the theory of Chow varieties, though it is a rather difficult moduli problem. It was defined by [Cay62] for curves in P 3. See Section 3.1 for an outline, [HP47] for a classical introduction and [Kol96, Secs.I.3 4] for a more recent treatment. Let k be an algebraically closed field and X a normal, projective k-variety. Fix a natural number m. An m-cycle on X is a finite, formal linear combination a i Z i where the Z i are irreducible, reduced subvarieties of dimension m and a i Z. We usually assume tacitly that all the Z i are distinct. An m-cycle is called effective if a i 0 for every i. Let Y X be a closed subscheme of dimension m. Let Y i Y be its m- dimensional irreducible components, Z i := red Y i and y i Y i the generic point. Let a i be the length of the Artin ring O yi,y i. We define the fundamental cycle of Y as [Y ] := a i Z i. Thus the fundamental cycle ignores lower dimensional associated primes and from the m-dimensional components it keeps only the underlying reduced variety and the length at the generic points. It turns out that there is a k-variety Chow m X, called the Chow variety of X whose points are in natural one-to-one correspondence with the set of effective m-cycles on X. Since we did not fix the degree of the cycles, Chow m X is not actually a variety but a countable disjoint union of projective, reduced k-schemes.

7 1.1. SHORT HISTORY OF MODULI PROBLEMS 7 The point of Chow m X corresponding to a cycle Z = a i Z i is also usually denoted by [Z]. As for linear systems, it is best to describe the natural correspondence by a universal family. The situation is, however, more complicated than before. There is a family or rather an effective cycle Univ m X on Chow m X X with projection π : Univ m X Chow m X such that for every effective m-cycle Z = a i Z i, 1 the support of π 1 [Z] is Z i, and 2 the fundamental cycle of u 1 [Z] equals Z if a i = 1 for every i. If the characteristic of k is 0, then the only problem in 2 is a clash between the traditional cycle-theoretic definition of the Chow variety and the scheme-theoretic definition of the fiber. It is easy to define a cycle-theoretic notion of fiber that restores equality in 2 for every Z; see [Kol96, I.3]. In positive characteristic the situation is more problematic; a possible solution is described in [Kol96, I.4]. The example of a perfect moduli problem is the theory of Hilbert schemes, introduced in [Gro62b]. See [Mum66], [Kol96, I.1 2] or [Ser06, Sec.4.3] for detailed treatments and Section 3.1 for a summary. 1.4 Hilbert schemes. Let k be an algebraically closed field and X a projective k-scheme. Set HilbX = {closed subschemes of X}. Then there is a k-scheme HilbX, called the Hilbert scheme of X whose points are in a natural one-to-one correspondence with closed subschemes of X. The point of HilbX corresponding to a subscheme Y X is frequently denoted by [Y ]. There is a universal family UnivX HilbX X such that 1 the first projection π : UnivX HilbX is flat, and 2 π 1 [Y ] = Y for every closed subscheme Y X. The beauty of the Hilbert scheme is that it describes not just subschemes but all flat families of subschemes as well. To see what this means, note that for any morphism g : T HilbX, by pull-back we obtain a flat family of subschemes of X parametrized by T T g,hilbx UnivX T X. It turns out that every family is obtained this way: 3 For every T and for every closed subscheme Z T T X that is flat and proper over T, there is a unique g : T HilbX such that Z T = T g,hilbx UnivX. This takes us to the next, functorial approach to moduli problems. 1.5 Hilbert functor and Hilbert scheme. Let X S be a morphism of schemes. Define the Hilbert functor of X/S as a functor that associates to a scheme T S the set Hilb X /S T = { subschemes Z T S X that are flat and proper over T }. The basic existence theorem of Hilbert schemes then says that, if X S is quasiprojective, there is a scheme Hilb X/S such that for any S scheme T, Hilb X /S T = Mor S T, HilbX /S.

8 8 1. INTRODUCTION Moreover, there is a universal family π : Univ X/S Hilb X/S such that the above isomorphism is given by pulling back the universal family. We can summarize these results as follows Principle 1.6. π : Univ X/S Hilb X/S contains all the information about proper, flat families of subschemes of X/S and does it in the most succinct way. This example leads us to a general definition: Definition 1.7 Fine moduli spaces. Let V be a reasonable class of projective varieties or schemes, or sheaves, or... In practice reasonable may mean several restrictions, but for the definition we only need the following weak assumption: 1 Let K k be a field extension. Then a k-variety X k is in V iff X K := X k Spec k Spec K is in V. Following 1.5, define the corresponding moduli functor as Flat families X T such that Varieties V T := every fiber is in V, modulo isomorphisms over T. We say that a scheme Moduli V, or, more precisely, a flat morphism u : Univ V Moduli V is a fine moduli space for the functor Varieties V if the following holds: 3 For every scheme T, pulling back gives an equality Varieties V T = Mor T, Moduli V. Applying the definition to T = Spec K, where K is a field, we see that every fiber of u : Univ V Moduli V is in V and the K-points of the fine moduli space Moduli V are in one-to-one correspondence with the K-isomorphism classes of objects in V. We consider the existence of a fine moduli space as the ideal possibility. Unfortunately, it is rarely achieved. 1.8 Remarks on flatness. The definition 1.7 is very natural within our usual framework of algebraic geometry, but it hides a very strong supposition: Assumption If V is a reasonable class then any flat family whose fibers are in V is a reasonable family. In Grothendieck s foundations of algebraic geometry flatness is one of the cornerstones and there are many reasonable classes for which flat families are indeed the reasonable families. Nonetheless, should not be viewed as self evident. Even when the base of the family is a smooth curve, needs arguing, but the assumption is especially surprising when applied to families over non-reduced schemes T. Consider, for instance, the case when T is the spectrum of an Artinian k-algebra. Then T has only one closed point t T. A flat family p : X T has only one fiber X t, and our only restriction is that X t be in our class V. Thus declares that we care only about X t. Once X t is in V, every flat deformation of X t over T is automatically reasonable. A crucial conceptual point in the moduli theory of higher dimensional varieties is the realization that in 1.7 flatness of the map X T is not enough: allowing

9 1.1. SHORT HISTORY OF MODULI PROBLEMS 9 all flat families whose fibers are in a reasonable class leads to the wrong moduli problem. Problems arise even for families of surfaces over smooth curves. The difficulty of working out the correct concept has been one of the main stumbling blocks of the general theory. Next we see what happens with the simplest case, for smooth curves of fixed genus. 1.9 Moduli functor and moduli space of smooth curves. Following 1.7 we define the moduli functor of smooth curves of genus g as Smooth, proper families S T, Curves g T := every fiber is a curve of genus g, modulo isomorphisms over T. It turns out that there is no fine moduli space for curves of genus g. Every curve C with nontrivial automorphisms causes problems; there can not be any point [C] corresponding to it in a fine moduli space. Actually, problems arise already when V consist of a single curve! See Section 1.7 for such examples. It has been, however, understood for a long time that there is some kind of an object, denoted by M g, and called the coarse moduli space or simply moduli space of curves of genus g that comes close to being a fine moduli space: 1 For any algebraically closed field k, the k-points of M g are in a natural one-to-one correspondence with isomorphism classes of smooth curves of genus g defined over k. Let us denote the correspondence by C [C] M g. 2 For any family of smooth genus g curves h : S T there is a moduli map m h,t : T M g such that for every geometric point p T, the image m h,t p is the point corresponding to the fiber [h 1 p]. For elliptic curves we get M 1 = A 1 and the moduli map is given by the j- invariant, as was known to Euler and Lagrange. They also knew that there is no universal family over M 1. The theory of Abelian integrals due to Abel, Jacobi and Riemann does essentially the same for all curves, though in this case a clear moduli theoretic interpretation seems to have been done only later [?]. For smooth plane curves, and more generally for smooth hypersurfaces in any dimension, the invariant theory of Hilbert produces coarse moduli spaces. Still, a precise definition and proof of existence of M g appeared only in [Tei44] in the analytic case and in [Mum65] in the algebraic case. See [AJP16] for a historical account Coarse moduli spaces. As in 1.7, let V be a reasonable class. When there is no fine moduli space, we still can ask for a scheme that best approximates its properties. We look for schemes M for which there is a natural transformation of functors T M : Varieties g Mor, M. Such schemes certainly exist, for instance, if we work over a field k then we can take M = Spec k. All schemes M for which T M exists form an inverse system which is closed under fiber products. Thus, as long as we are not unlucky, there is a universal or largest scheme with this property. Though it is not usually done, it should be called the categorical moduli space.

10 10 1. INTRODUCTION This object can be rather useless in general. For instance, fix n, d and let H n,d be the class of all hypersurfaces of degree d in P n+1 k up to isomorphisms. It is easy to see cf that a categorical moduli space exists and it is Spec k. To get something more like a fine moduli space, we require that it give a one-toone parametrization, at least set theoretically. Thus we say that a scheme Moduli V is a coarse moduli space for V if the following hold. 1 There is a natural transformation of functors ModMap : Varieties V Mor, Moduli V, 2 Moduli V is universal satisfying 1, and 3 for any algebraically closed field K k, ModMap : Varieties V Spec K = MorSpec K, Moduli V = Moduli V K is an isomorphism of sets Moduli functors versus moduli spaces. While much of the early work on moduli, especially since [Mum65], put the emphasis on the construction of fine or coarse moduli spaces, recently the emphasis shifted towards the study of the families of varieties, that is towards moduli functors and moduli stacks. The main task is to understand what kind of objects form nice families. Once a good concept of nice families is established, the existence of a coarse moduli space should be nearly automatic. The coarse moduli space is not the fundamental object any longer, rather it is only a convenient way to keep track of certain information that is only latent in the moduli functor or moduli stack Compactifying M g. While the basic theory of algebraic geometry is local, that is, it concerns affine varieties, most really interesting and important objects in algebraic geometry and its applications are global, that is, projective or at least proper. The moduli spaces M g are not compact, in fact the moduli functor of smooth curves discussed so far has a definitely local flavor. Most naturally occurring smooth families of curves live over affine schemes, and it is not obvious how to write down any family of smooth curves over a projective base. For many reasons it is useful to find geometrically meaningful compactifications of M g. The answer to this situation is to allow not just smooth curves but also singular curves in our families. Concentrating on 1-parameter families, the main question is the following: Let B be a smooth curve, B 0 B an open subset and π 0 : S 0 B 0 a smooth family of genus g curves. Find a natural extension S 0 S π 0 π B 0 B, where π : S B is a flat family of possibly singular curves. We would like the extension to be unique and behave well with respect to pulling back families over curves and for families over higher dimensional bases. The answer, proposed in [DM69] has been so successful that it is hard to imagine a time when it was not the obvious solution. Let us first review the definition of [DM69]. In Section 1.4 we see, by examples, why this concept has not been so obvious.

11 1.1. SHORT HISTORY OF MODULI PROBLEMS 11 Definition 1.13 Stable curve. A stable curve over an algebraically closed field k is a proper, connected k-curve C such that the following hold: Local property The only singularities of C are ordinary nodes. Global property The canonical or dualizing sheaf ω C is ample. A stable curve over a scheme T is a flat, proper morphism π : S T such that every geometric fiber of π is a stable curve. The arithmetic genus of the fibers is a locally constant function on T, but we usually also tacitly assume that it is constant. The moduli functor of stable curves of genus g is { } Stable curves of genus g over T, Curves g T := modulo isomorphisms over T. Theorem [DM69] For every g 2, the moduli functor of stable curves of genus g has a coarse moduli space M g. Moreover, Mg is projective, normal, has only quotient singularities and contains M g as an open dense subset. M g has a rich and intriguing intrinsic geometry which is related to major questions in many branches of mathematics and theoretical physics; see [FM13] for a collection of surveys Moduli for varieties of general type. The aim of this book is to use the moduli of stable curves as guideline, and develop a moduli theory for varieties of general type. For the non-general type case, see In some sense, this is a hopeless task since higher dimensional varieties are much more complicated than curves. For instance, even for smooth surfaces with ample canonical class, the moduli spaces can have arbitrarily complicated singularities and scheme structures [Vak06]. Thus we approach the question in four stages: 1 Develop the correct higher dimensional analog of smooth, projective curves of genus 2. 2 Following the example of stable curves, define the notion of stable varieties in higher dimensions. 3 Show that the functor of stable varieties with suitably fixed numerical invariants gives a well behaved moduli functor/stack and has a projective coarse moduli space. 4 Show that, in many important cases, these moduli spaces are interesting and useful objects. Let us now see in some detail how these goals are accomplished Higher dimensional analogs of smooth curves of genus 2. It has been understood since the beginnings of the theory of surfaces that, for surfaces of Kodaira dimension 0, the correct moduli theory should be birational, not biregular. That is, the points of the moduli space should correspond not to isomorphism classes of surfaces but to birational equivalence classes of surfaces. There are two ways to deal with this problem. First, one can work with smooth families but consider two families equivalent of there is a rational map between them that induces a birational equivalence on every fiber. This seems rather complicated technically. The second, much more useful method relies on the observation that every birational equivalence class of surfaces of Kodaira dimension 0 contains a unique

12 12 1. INTRODUCTION minimal model, that is, a smooth projective surface S m whose canonical class is nef. Therefore, one can work with families of minimal models, modulo isomorphisms. With the works of [Mum65, Art74] it became clear that, for surfaces of general type, it is even better to work with the canonical model, which is a mildly singular projective surface S c whose canonical class is ample. The resulting class of singularities has been since established in all dimensions; they are called canonical singularities See Section 1.2 for details. Principle In moduli theory, the main objects of study are projective varieties with ample canonical class and with canonical singularities. The correct definition of the higher dimensional analogs of stable curves was much less clear. An approach through geometric invariant theory was investigated [Mum77], but never fully developed. In essence, the GIT approach starts with a particular method of construction of moduli spaces and then tries to see for which class of varieties does it work. The examples of [WX14] suggest that geometric invariant theory is unlikely to give a good compactification for the moduli of surfaces. A different framework was proposed in [KSB88]; see also [Ale96]. Instead of building on geometric invariant theory, it focuses on 1-parameter families and uses Mori s program as its basic tool. Before we give the definition, it is very helpful to go through a key step of the proof of 1.14 that establishes separatedness and properness of Mg. Keeping in mind the valuative criteria of separatedness and properness , we expect the difficulties to be essentially 1-dimensional. This is the topic of the next theorem. Theorem 1.17 Stable reduction for curves. Let B be a smooth curve, B 0 B an open subset and π 0 : S 0 B 0 a flat family of genus g stable curves. Then there is a finite surjection p : A B such that there is a unique extension S 0 B A =: T 0 T π 0 A π A B 0 B A =: A 0 A, where π A : T A is a flat family of genus g stable curves Outline of proof of Let us present the process in a way that generalizes to higher dimensions. Main case The generic fiber of π 0 : S 0 B 0 is smooth. Step 1.1. Take any possibly singular projective surface S 1 S 0 such that π 0 extends to a morphism π 1 : S 1 B. Step 1.2. Resolve the singularities of S 1 to obtain a smooth surface π 2 : S 2 B such that the reduced fibers of π 2 have only nodes as singularities. Step 1.3. Run the relative minimal model program. That is, repeatedly contract all smooth rational curves C S 2 that are contained in a fiber of π 2 and have negative intersection with the canonical class. The end result is π 3 : S 3 B where K S3 has non-negative degree on all curves contained in any fiber of π 3. Step 1.4. Take the relative canonical model. That is, contract all smooth rational curves C S 3 that are contained in a fiber of π 3 and have zero intersection with the canonical class. The end result is π 4 : S 4 B where K S4 has positive degree on all curves contained in any fiber of π 3. Thus K S4 is relatively ample.

13 1.1. SHORT HISTORY OF MODULI PROBLEMS 13 Note that S 4 is, in general, not smooth, but has very simple so called Du Val singularities. Step 1.5. Prove that π 4 : S 4 B is the unique surface containing S 0 that has Du Val singularities and relatively ample canonical class. Step 1.6. In general, the fibers of π 4 are not reduced and the construction of S 4 does not commute with base change p : A B. However, if the fibers of π 2 are reduced, then the fibers of π 4 are stable curves and the construction of S 4 does commute with base change. Assuming only that the fibers of π 4 be reduced would not be enough. Step 1.7. Show that if p : A B is sufficiently ramified and T 0 := S 0 B A then the analogously constructed T := T 4 A satisfies the requirements of Just to be concrete, in characteristic 0, the following ramification condition is sufficient: For every a A, the ramification index of p at a is divisible by the multiplicity of every irreducible component of π2 1 pa. Secondary case The generic fiber of π 0 : S 0 B 0 is not normal. Step 2.0. The generic fiber of π 0 : S 0 B 0 has nodes, and, correspondingly, S 0 has normal crossing singularities along a curve C 0 S 0. Let S 0 S 0 be the normalization and D 0 S 0 the preimage of the double curve. We also keep track of the involution τ 0 of the degree 2 cover D 0 C 0. Steps Run the analog of Steps for S 0 B 0, with the difference of using canonical class + birational transform of D 0 everywhere instead of the canonical class. The end result is π T : T A with D T T the curve corresponding to D 0. Step 2.8. Show that the involution τ 0 extends to an involution τ T on D T. Construct a new, non-normal surface σ : T T such that σ is an isomorphism outside D T and we identify every point p D T with its image τ T p Higher dimensional analogs of stable curves of genus 2. Now we can state the main theses of [KSB88] about higher dimensional moduli problems: Principle In higher dimensions, we should follow the proof of the Stable reduction theorem 1.17 as outlined in The resulting fibers give the right class of stable varieties. Principle As in 1.13, a connected k-scheme X is stable iff it satisfies the following two conditions: Local property A restriction on the singularities of X so-called semi-logcanonical singularities. Global property The canonical or dualizing sheaf ω X is ample. The definition of semi-log-canonical is not important for now 1.41, the key point is that the only global restriction is the ampleness of ω X. In general, Step 1.1 of 1.18 is still easy and Step 1.2 uses Hironaka s resolution of singularities. Steps use Mori s program, also called the minimal model program. When [KSB88] was written, the relevant results were only known for families of surfaces, but [BCHM10] and [HX13] take care of the higher dimensional cases as well.

14 14 1. INTRODUCTION Steps need very little change. As a starting point one could use the Semistable reduction theorem [KKMSD73], but, as we see in Section 2.4, one can get by without it. Steps of the secondary case have not been worked out earlier. Steps mostly work as before; the relevant results of the minimal model program have been established in [HX13]. Step 2.8 turned out to be unexpectedly subtle. It is closely related to some basic questions concerning semi-log-canonical schemes. These were settled in [Kol16b] and a detailed treatment was given in [Kol13c, Chap.5]. An alternative way to approach this case would be to develop the minimal model program for varieties with normal crossing singularities and apply it directly, without normalizing in Step 2.0. Much of the background for such an approach is worked out in [Fuj14]. However, it turns out that the minimal model program fails already for surfaces with normal crossing singularities [Kol11c] Moduli functor of stable varieties. In the moduli theory of curves, we go directly from the definition of stable curves over fields to the notion of stable curves over an arbitrary base By contrast, for surfaces and in higher dimensions, a major difficulty remains. As we already mentioned in 1.8, not every flat family of stable surfaces can be allowed in a reasonable moduli theory. Examples illustrating this are given in Section 1.3. We must restrict to families S T where the Hilbert function of the fibers χ S t, O St mk St is independent of t T. The problem is that, for stable varieties, the canonical class K need not be Cartier, and the sheaves O St mk St do not form a flat family over T. It is actually quite difficult to define the right concept. Our final solution of this problem is in Chapter??? Good properties of moduli problems. Let V be a reasonable class of varieties and Varieties V the corresponding moduli functor. It is hard to pin down exactly what reasonable should mean, but it seems nearly impossible to do anything without the following assumption: Representability The functor Varieties V is representable by a monomorphism 3.47 if for any flat morphism X S there is a monomorphism S V S such that for any g : T S, the pull-back X S T T is in Varieties V T iff g factors as g : T S V S. In many cases, S V S is an open embedding. For instance, being reduced, normal or smooth are all open conditions. On the other hand, being a hyperelliptic curve is not an open condition but it is a locally closed condition. Representability also implies that membership in Varieties V T can be tested on 0-dimensional subschemes of T, that is, on spectra of Artin rings. This is the reason why formal deformation theory is such a powerful tool [Ill71, Art76, Ser06]. Assume for the moment that there is a coarse moduli space Moduli V. Our next aim is to understand how to recognize properties of Moduli V in terms of the functor Varieties V. Let X be a scheme of finite type over a field k. By the valuative criterion of separatedness, X is separated iff the following holds.

15 1.1. SHORT HISTORY OF MODULI PROBLEMS 15 Let B be a smooth curve over k and B 0 B an open subset. Then a morphism τ 0 : B 0 X has at most one extension to τ : B X. If X = Moduli V is a fine moduli space, then giving a morphism U X is equivalent to specifying a proper, flat family V U U whose fibers are in V. Thus the valuative criterion of separatedness translates to functors as follows: Separatedness The functor Varieties V is separated iff for every smooth curve B and every open subset B 0 B, a proper, flat family π 0 : V 0 B 0 whose fibers are in V has at most one extension to V 0 V π 0 π B 0 B, where π : V B is also a proper, flat family whose fibers are in V. We obtain a similar translation of the valuative criterion of properness, but here we have to pay attention to the difference between coarse and fine moduli spaces. Valuative criterion of properness The functor Varieties V satisfies the valuative criterion of properness iff the following holds: Let B be a smooth curve, B 0 B an open subset and π 0 : V 0 B 0 a proper, flat family whose fibers are in V. Then there is a finite surjection p : A B such that there is an extension V 0 B A =: W 0 W π A B 0 B A =: A 0 A, where π A : W A is also a proper, flat family whose fibers are in V. For functors with a fine moduli space, we could take A = B, but otherwise a finite base change may be needed. It is very convenient to roll these two concepts together. The resulting condition is then exactly the general version of the Stable reduction theorem The valuative criterion of properness implies properness for schemes of finite type, but not in general. The next condition is the functor version of finite type. It ensures that we do not have too many objects to parametrize. Boundedness The class of schemes V is called bounded if there is a flat morphism of schemes of finite type u : U T such that for every algebraically closed field K, every K-scheme in V occurs as a fiber of U K T K. Some authors also assume that every fiber of u : U T is in V. How important are these conditions? As we already noted, the assumption in this book is that representability is indispensable. When separatedness fails, it usually either fails very badly or it can be restored by a judicious change of the definition; see Section 1.4 for such examples. Note, however, that most moduli functors of sheaves behave differently. They are not separated but the notions of semi-stability and GIT quotients provide a good method to deal with this. See [HL97] for details.

16 16 1. INTRODUCTION Properness is considered a challenge: If a moduli functor does not satisfy the valuative criterion of properness, we should try to enlarge the moduli problem to a proper one. Finally, boundedness seems to come automatically, though it can be very hard to prove that it holds. I do not know any natural moduli functor of projective varieties satisfying with a coarse moduli space whose connected components are not of finite type. In the proper but non-projective setting this can, however, happen. The Hilbert scheme of curves on the Hironaka 3-fold described in [Har77, App.B.3.4.1] has a connected component with infinitely many irreducible components, each proper. I do not know any natural moduli functor with a coarse moduli space that has an irreducible component that is not of finite type From the moduli functor to the moduli space. Starting with [Mum65] and [Mat64], much effort was devoted to going from the moduli functor Varieties V to the moduli space Moduli V. In the quasi-projective setting, this was solved in [Vie95], but the proofs are quite hard. The construction of the moduli space as an algebraic space turns out to be much easier, and the general quotient theorems of [Kol97, KM97] take care of it completely, see also [Ols16]. Once we have a moduli space which is a proper algebraic space, one needs to prove that it is projective. For surfaces this was done in [Kol90] and extended to higher dimensions in [Fuj12] and [KP17] Moduli for varieties of non-general type. In contrast with varieties of general type, the moduli theory for varieties of non-general type is very complicated. A general problem, illustrated by Abelian, elliptic and K3 surfaces is that a typical deformation of such an algebraic surface over C is a non-algebraic complex analytic surface. Thus any algebraic theory captures only a small part of the full analytic deformation theory. The moduli question for analytic surfaces has been studied, especially for complex tori and K3 surfaces. In both cases it seems that one needs to add some extra structure for instance, fixing a basis in some topological homology group in order to get a sensible moduli space. As an example of what could happen, note that the 3-dimensional space of Kummer surfaces is dense in the 20-dimensional space of all K3 surfaces, cf. [PŠŠ71]. Even if one restricts to the algebraic case, compactifying the moduli space seems rather hopeless. Detailed studies of Abelian varieties and K3 surfaces show that there are many different compactifications depending on additional choices, see [KKMSD73, AMRT75]. It is only with the works of [Ale02] that a geometrically meaningful compactification of the moduli of principally polarized Abelian varieties became available. This relies on the observation that a pair A, Θ consisting of a principally polarized Abelian variety A and its theta divisor Θ behaves as if it were a variety of general type Further problems. While we provide a solution to the basic general questions of the moduli theory of varieties of general types, there are many unsolved aspects. Some of the main ones are the following.

17 1.2. FROM SMOOTH CURVES TO CANONICAL MODELS 17 Problem Positive characteristic. Most of our results work only in characteristic 0. This is partly caused by the need for resolution of singularities and minimal model theory. There are, however, many other difficulties that are unsettled in positive characteristic. Even the correct definition of stable families is problematic The paper [Pat14] makes substantial progress on this. Problem Boundedness. We show that in our moduli spaces, every irreducible component is projective. It is much harder to rule out the possibility of a connected component with infinitely many irreducible components. A solution of this question follows from the deep results of [HMX14] that cover a series of interesting conjectures on various numerical invariants satisfying the ascending chain condition. A simpler proof would be very desirable. Problem Effective results. Given a class of varieties of general type, we do not have good general methods to decide which stable varieties occur on the corresponding components of the moduli space. Even bounding basic numerical invariants, for instance the number of irreducible components, seems very hard. The methods in Section??? provide an answer in principle, but it does not seem feasible to work it out in practice, save in some very simple cases. A few results are discussed in Section???, but it would be very useful to get much more information. Problem Fine moduli spaces. As we see, stable varieties have finite automorphism groups, and we get a fine moduli space iff the identity is the only automorphism; see Section 1.7. Hence the question: Is there a sensible way to kill automorphisms by additional structures. For curves over C this is achieved by introducing a level m structure for some m 3, that is, by fixing an isomorphism H 1 C, Z/m = Z/m 2g. For smooth surfaces, similar topological invariants do not seem to be sufficient, but a completely different approach may work. Problem Applications. Many basic questions about smooth curves can be solved by investigating an analogous problem on stable curves, whose geometry is frequently much simpler. There are, so far, few such results in higher dimensions. Some of these are discussed in Section???. For example, [LP07, PPS09a, PPS09b] use stable surfaces to construct new examples of smooth surfaces and 4-manifolds. One problem is that it is not easy to write down stable degenerations, the other is that the stable varieties themselves are rather complicated From smooth curves to canonical models In the theory of curves, the basic objects are smooth projective curves. We frequently study any other curve by relating it to smooth projective curves. This is why the moduli functor/space of smooth curves is so important. In higher dimensions, we define the moduli functor of smooth varieties as { } Smooth, proper families X S, SmoothS := modulo isomorphisms over S. This, however, gives a rather badly behaved and mostly useless moduli functor already for surfaces. First of all, it is very non-separated Non-separatedness in the moduli of smooth surfaces of general type. We construct two smooth families of projective surfaces f i : X i B over a pointed smooth curve b B such that

18 18 1. INTRODUCTION 1 all the fibers are smooth, projective surfaces of general type, 2 X 1 B and X 2 B are isomorphic over B \ {b}, 3 the fibers Xb 1 and X2 b are not isomorphic. As the construction shows, this type of behavior happens every time we look at deformations of a surface that contains at least three 1-curves. Let f : X B be a smooth family of projective surfaces over a smooth affine pointed curve b B. Let C 1, C 2, C 3 X be three sections of f, all passing through a point x b X b with independent tangent directions and are disjoint elsewhere. Set X 1 := B C1 B C2 B C3 X, where we first blow-up C 3 X, then the birational transform of C 2 in B C3 X and finally the birational transform of C 1 in B C2 B C3 X. Similarly, set X 2 := B C1 B C3 B C2 X. Since the C i are sections, all these blow-ups are smooth families of projective surfaces over B. Over B \ {b} the curves C i are disjoint, thus X 1 and X 2 are both isomorphic to B C1+C 2+C 3 X, the blow-up of C 1 + C 2 + C 3 X. We claim that, by contrast, the fibers of Xb 1 and X2 b are not isomorphic to each other for a general choice of the C i. To see this, choose local analytic coordinates t at b B and x, y, t at x b X. The curves C i are defined by equations C i = x a i t higher terms = y b i t higher terms = 0. The blow-up B Ci X is given by B Ci X = u i x a i t higher terms = v i y b i t higher terms X P 1 u iv i. On the fiber over b these give the same blow-up B xb Xb = ux = vy Xb P 1 uv Thus we see that the birational transform of C j intersects the central fiber B Ci X = b B xb Xb at the point u v = a j a i {x b } P 1 b j b uv. i The fibers B C2 B C3 X and B b C3 B C2 X are isomorphic to each other since they b are obtained from B xb Xb by blowing up the same point u v = a 2 a 3 b 2 b 3 resp. u v = a 3 a 2 b 3 b 2. When we next blow up the birational transform of C 1 on B C2 B C3 X resp. on b BC3 B C2 X this gives the blow-up of the point b a 1 a 3 a 1 a 2 resp., b 1 b 3 b 1 b 2 and these are different, unless C 1 + C 2 + C 3 is locally planar at x b. So far we have seen that the identity X b = X b does not extend to an isomorphism between the fibers Xb 1 and X2 b. If X b is of general type, then Aut X b is finite, hence, to ensure that Xb 1 and X2 b are not isomorphic, we need to avoid finitely many other possible coincidences in The main reason, however, why we do not study the moduli functor of smooth varieties up to isomorphism is that, in dimension two, smooth projective surfaces do not form the smallest basic class. Given any smooth projective surface S, one

19 1.2. FROM SMOOTH CURVES TO CANONICAL MODELS 19 can blow up any set of points Z S to get another smooth projective surface B Z S which is very similar to S. Therefore, the basic object should be not a single smooth projective surface but a whole birational equivalence class of smooth projective surfaces. Thus it would be better to work with smooth, proper families X S modulo birational equivalence over S. That is, with the moduli functor GenType bir S := Smooth, proper families X S, every fiber is of general type, modulo birational equivalences over S In essence this is what we end up doing, but it is very cumbersome do deal with birational equivalence over a base scheme. Nonetheless, working with birational equivalence classes leads to a separated moduli functor. Proposition Let f i : X i B be two smooth families of projective varieties over a smooth curve B. Assume that the generic fibers XkB 1 and X2 kb are birational and the pluricanonical system mk X 1 is nonempty for some m > 0. Then, for every b B, the fibers X 1 b and X2 b kb are birational. Proof. Pick a birational map φ : XkB 1 X2 kb and let Γ X1 B X 2 be the closure of the graph of φ. Let Y Γ be the normalization with projections p i : Y X i. Note that both of the p i are open embeddings on Y \ Ex p 1 Ex p 2. Thus if we prove that neither p 1 Ex p1 Ex p 2 nor p2 Ex p1 Ex p 2 contains a fiber of f 1 or f 2, then p 2 p 1 1 : X 1 X 2 restricts to a birational map Xb 1 X2 b for every b B. Thus the fiber Y b contains an irreducible component that is the graph of the birational map Xb 1 X2 b, but it may have other components too; see We use the canonical class to compare Ex p 1 and Ex p 2. Since the X i are smooth, K Y p i K X i + E i, where E i 0 and Supp E i = Ex p i Assume for simplicity that B is affine and let Bs mk X i denote the set-theoretic base locus. By assumption, mkx i is not empty and since B is affine, Bs mkx i does not contain any of the fibers of f i. Every section of OmK Y pulls back from X i, thus Bs mk Y = p 1 i Bs mk X i + Supp E i. Comparing these for i = 1, 2, we conclude that p 1 1 Bs mk X 1 + Supp E 1 = p 1 2 Bs mk X 2 + Supp E 2. Therefore, p 1 Supp E2 p1 Supp E1 + Bs mkx 1. Since E 1 is p 1 -exceptional, p 1 E1 has codimension 2 in X 1, hence it does not contain any of the fibers of f 1. We saw that Bs mkx 1 does not contain any of the fibers either. Thus p 1 Ex p1 Ex p 2 does not contain any of the fibers and similarly for p 2 Ex p1 Ex p 2. Remark A result of [MM64] says that, more generally, 1.26 holds as long as the fibers Xb i are not birationally ruled, that is, not birational to a variety of the form Z P 1. The proof of [MM64], relies on the study of exceptional divisors

20 20 1. INTRODUCTION over a smooth variety; see [KSC04, Sec.4.5] for an overview. Exceptional divisors over a singular variety are much less understood. By contrast, the above proof focuses on the role of the canonical class. It is worthwhile to go back and check that the proof works if the X i are normal, as long as holds; the latter is essentially the definition of terminal singularities. It is precisely the property and its closely related variants that lead us to the correct class of singular varieties for moduli purposes. Since it is much harder to work with a whole equivalence class, it would be desirable to find a particularly nice surface in every birational equivalence class. This is achieved by the theory of minimal models of algebraic surfaces. By a result of Enriques cf. [BPV84, III.4.5], every birational equivalence class of surfaces S contains a unique smooth projective surface whose canonical class is nef that is, has nonnegative degree on every effective curve, except when S contains a ruled surface C P 1 for some curve C. This unique surface is called the minimal model of S. It would seem at first sight that 1.26 implies that the moduli functor of minimal models is separated. There is, however, a quite subtle problem Non-separatedness in the moduli of minimal models. We construct two smooth families of projective surfaces f i : X i B over a pointed smooth curve b B such that 1 all the fibers are smooth, projective minimal models, 2 X 1 B and X 2 B are isomorphic over B \ {b}, 3 the fibers Xb 1 and X2 b are isomorphic, but 4 X 1 B and X 2 B are not isomorphic. While it is not clear from our construction, similar problems happen for any smooth family of surfaces where the general fiber has ample canonical class and a special fiber has nef but not ample canonical class, see [Art74, Bri68b, Rei80]. Let X 0 := fx 1,..., x 4 = 0 P 3 be a surface of degree n that has an ordinary double point at p = 0:0:0:1 as its sole singularity and contains the pair of lines x 1 x 2 = x 3 = 0. Let g be homogeneous of degree n 1 such that x n 1 4 appears in it with nonzero coefficient. Consider the family of surfaces X := fx 1,..., x 4 + tx 3 gx 1,..., x 4 = 0 P 3 x A 1 t. Note that X t is smooth for general t 0 and X contains the pair of smooth surfaces x 1 x 2 = x 3 = 0. For i = 1, 2, let X i := B xi,x 3X denote the blow-up of x i = x 3 = 0 with induced morphisms π i : X i X and f i : X i A 1. There is a natural birational map φ := π2 1 π 1 : X 1 X 2. Let B p X denote the blow-up of p = 0:0:0:1, 0 with exceptional divisor E B p X. We claim that the following hold. 5 The f i : X i A 1 are projective families of surfaces which are smooth over a neighborhood of t = 0. 6 For n 5, the fibers X i t have ample canonical class for t 0 and nef canonical class for t = 0. 7 X 1 X X 2 is isomorphic to B p X, hence it is smooth and irreducible. 8 The map φ is an isomorphism over A 1 \ {0} but it is not an isomorphism over 0.

21 1.2. FROM SMOOTH CURVES TO CANONICAL MODELS 21 9 The fiber of X 1 X X 2 over t = 0 has two irreducible components. One of these components is the graph of an isomorphism X 1 0 = X 2 0. The other component is E = P 1 P Thus φ : X 1 X 2 is an isomorphism over A 1 \ {0}, the X i A 1 have isomorphic fibers over 0 A 1, but φ is not an isomorphism over A 1. It is not hard to see that, for general choice of f and g, the X t have no birational self-maps, thus the only possible isomorphism between X 1 and X 2 would be the identity on X. Thus, by 6, in this case, X 1 and X 2 are not isomorphic to each other. Note that x i = x 3 = 0 defines a Weil divisor in X which is Cartier outside the point p. Thus all 3 blow-ups are isomorphisms over X \ {p}. This means that all the above claims are local near p. We prove the claims in after choosing better local coordinates near p that make all the assertions 5 10 transparent. All such problems go away when the canonical class is ample. Proposition Let f i : X i B be two smooth families of projective varieties over a smooth curve B. Assume that the canonical classes K X i are f i - ample. Let φ : XkB 1 = XkB 2 be an isomorphism of the generic fibers. Then φ extends to an isomorphism Φ : X 1 = X 2. Proof. Let Γ X 1 B X 2 be the closure of the graph of φ. Let Y Γ be the normalization, with projections p i : Y X i and f : Y B. As in 1.26, we use the canonical class to compare the X i. Since the X i are smooth, K Y p i K X i + E i where E i is effective and p i -exceptional Since p i O Y me i = O X i for every m 0, we get that f i O X i mkx i = f i p i O Y mp i K X i = f i p i O Y mp i K X i + me i = f i p i O Y mky = = = f O Y mky. Since the K X i are f i -ample, X i = Proj B m 0 f i O X i mkx i. Putting these together, we get the isomorphism Φ : X 1 = Proj B m 0 f 1 O X 1 mkx 1 = = Proj B m 0 f O Y mky = = Proj B m 0 f 2 O X 2 mkx 2 = X 2. Remark As in 1.27, it is again worthwhile to investigate the precise assumptions behind the proof. The smoothness of the X i is used only through the pull-back formula , which is weaker than If holds, then, even if the K X i are not f i -ample, we obtain an isomorphism Proj B f 1 O X 1 mkx 1 = ProjB f 2 O X 2 mkx m 0 Thus it is of interest to study objects as in in general. Let us start with the absolute case, when X is a smooth projective variety over a field k. Its canonical ring is the graded ring m 0 RX, K X := m 0 H 0 X, O X mk X.